21 problems

Let $\mathcal{K}$ be a knot, let $J_\mathcal{K}(N;q)$ denote its $N$-colored Jones polynomial, and let $A_\mathcal{K}(\ell,m)$ be the A-polynomial of the knot. The meridian and lon…

Let $K\subseteq S^3$ be a knot. Let $\hat{A}_K(t,M,L)$ be the quantum A-polynomial and let $A_K(M,L)$ denote the corresponding classical A-polynomial. AJ conjecture. The classical…

For a knot $\mathcal{K}$, let $\mathcal{H}_n(\mathcal{K};q,a)$ be the colored HOMFLY-PT invariant in the $n$th symmetric representation. Let $M$ and $L$ act by … with $LM=qML$. A n…

Let $K\subseteq S^3$ be a knot. Let $\hat A_K(t,M,L)$ be the noncommutative quantum A-polynomial obtained from the recursion ideal of the colored Jones polynomial, and let…

Let $K$ be a knot, let $P_r^K(a,q)$ denote its $r$-colored HOMFLY-PT polynomial, and let $\widehat{A}^K$ be the quantum $a$-deformed A-polynomial operator. A-polynomial recursion c…

AJ conjecture. For every knot $K\subset S^3$, ${\alpha}_K(1,\mathfrak{M},\mathfrak{L})$ is equal to $A_K(\mathfrak{M},\mathfrak{L})$ up to a factor depending only on $\mathfrak{M}$…

Character-variety symmetry conjecture. Condition (v) follows from conditions (i) and (ii). This conjecture is presented as a possible implication related to the hypotheses of the p…

Let $K\subset S^3$ be a knot. Denote by $\alpha_K(t,M,L)$ its recurrence polynomial and by $A_K(L,M)$ its $A$-polynomial. AJ conjecture. For every knot $K\subset S^3$, the speciali…

Let $K$) be a knot in the $3$-sphere, and let $calpha_K(t,M,L)$ be its recurrence polynomial. Let the $A$-polynomial of $K$ be the classical polynomial in the meridian and longitud…

A-polynomial augmentation conjecture. The $a$-deformed $A$-polynomial of $SU$-type should be identifiable with the augmentation polynomial of knot contact homology:

SO-type orientifold moduli-space conjecture. In the presence of an orientifold, the disc-corrected moduli space of the Lagrangian brane associated to $K$ should be the zero locus

Aganagic's moduli-space conjecture. The disc-corrected moduli space of $\mathcal L_K$ is given by

Let $K$ be a link, let its augmentation variety encode the relevant large-$N$ data, and let the colored HOMFLY polynomial be considered in the large-$N$ limit. Large-$N$ colored HO…

Let $K$ be a knot, and let $A^{\text{super}}(x,y;a,t)$ be its super-$A$-polynomial. A polynomial is quantizable when its Newton polygon has the required temperedness property, name…

Quantization conjecture for super-A-polynomials. The $Q$-deformed $A$-polynomial is identified with the augmentation polynomial of knot contact homology. This proposed identificati…

Generalized volume conjecture for colored superpolynomials. In this limit, the large-color behavior is

Let $A(K;x,y)$ be the classical $A$-polynomial of a knot $K$, and let $\widehat A(K;\hat x,\hat y;q)$ be its quantum version. Let $Z_{\mathrm{CS}}$ denote the Chern–Simons partitio…

Let $\mathfrak{M}$ be a hyperbolic $3$-manifold with one cusp, let $(\mathcal{C}_0,u,v)$ be an A-spectral curve from an irreducible component of its A-polynomial, and let…

Let $\mathfrak{M}$ be a hyperbolic $3$-manifold with A-polynomial $\mathfrak{A}(m,l)$, and let $\widehat{\mathfrak{A}}_{\hbar}(m,l,e^{2\hbar})$ be the corresponding quantum operato…

Let $\mathfrak{K}$ be a hyperbolic knot, let $J_N(q)$ be its colored Jones polynomial, and let $\mathcal{C}$ be the semiclassical spectral curve of an integrable system. All-order…

Let $K$ be a knot and let $A^{\text{ref}}(x,y;t)$ be its refined $A$-polynomial, with $t$ a continuous parameter. Root-of-unity quantizability conjecture. For any knot, quantizabil…